Some results about equichordal convex bodies
Jes\'us Jer\'onimo-Castro, Francisco G. Jimenez-Lopez, Efr\'en Morales-Amaya

TL;DR
This paper investigates the existence of equichordal bodies within convex bodies, showing that in higher dimensions only spheres have such bodies, and explores related geometric properties like isoptic curves.
Contribution
It proves the existence of non-circular equichordal bodies in planar convex bodies and establishes that only spheres have them in dimensions three and higher.
Findings
Multiple non-circular equichordal convex bodies exist in the plane.
In dimensions ≥3, only Euclidean balls have equichordal convex bodies.
Connections between isoptic curves and convex rotors are established.
Abstract
Let and be two convex bodies in , , with . We say that is an equichordal body for if every chord of tangent to has length equal to a given fixed value . J. Barker and D. Larman proved that if is a ball, then is a ball concentric with . In this paper we prove that there exist an infinite number of closed curves, different from circles, which possess an equichordal convex body. If the dimension of the space is more than or equal to 3, then only Euclidean balls possess an equichordal convex body. We also prove some results about isoptic curves and give relations between isoptic curves and convex rotors in the plane.
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Taxonomy
TopicsPoint processes and geometric inequalities · Connective tissue disorders research · Mathematics and Applications
